Realize the park is huge, and it is lot of miles from place to place.
Campgrounds - Big Bend National Park (U.S. National Park Service)
Boondocking camping can also be done at the Cottonwood campground on the west side along the river. No dump, no generators are allow.
Up high in the Basin, no hook ups, and a limit on trailer and RV lengths.
Rio Grande Village already mentioned, has the water, dump and 2nd place to find gas in the park. Boondocking, generator area, and then the parking lot hook up place.
Always check the weather. When it rains, you can be stuck and stranded on the off road routes. Much of the park has little or no cell service. We rode the dirt river park road, west to east, for 56 miles, saw, 1 moving vehicle, 2 parked, and had no cell service on much of the route.
Outside the park, on the east side, down the deadend highway 2627, off 385, is Stillwell Store. Stillwell Store
Outside the park on the west side are more choices as well.
Look at the distance from Alpine to the west park entrance, is some 80 miles. From the park's west entrance up to a star party and back is 242 miles, in the dark in deer and javelina country.
I would second driving the route to Precidio and back on 170. But know it is 242 miles round trip from Rio Grande Village. But plan to do lunch in Precidio, and take your camera!